Third order solutions of cosmological density perturbations in Horndeski’s most general scalar-tensor theory with the Vainshtein mechanism

We study the third order solutions of the cosmological density perturbations in Horndeski’s most general scalar-tensor theory under the condition that the Vainshtein mechanism is at work. In this work, we thoroughly investigate the independence property of the functions describing the nonlinear mode couplings, which is also useful for models within general relativity. Then, we find that the solutions of the density contrast and the velocity divergence up to third order are characterized by six parameters. Furthermore, the one-loop order power spectra obtained with third order solutions are described by four parameters. We exemplify the behavior of the one-loop order power spectra assuming the kinetic gravity braiding model, which demonstrates that the effect of modified gravity appears more significantly in the power spectrum of the velocity divergence than the density contrast.

Figure 1

$P_{\delta \delta }(t,k)$ (thick curve) and the corresponding linear power spectrum $D_{+}^2(t)P_L(k)$ (thin curve) at the redshift $z=0$ (solid curve), $z=1$ (dashed curve), and $z=1.5$ (dotted curve), respectively, for the $\mathrm{\Lambda }\mathrm{CDM}$ model with the cosmological parameters ${\mathrm{\Omega }}_0=0.26$, ${\mathrm{\Omega }}_b=0.044$, $h=0.72$, $n_s=0.96$, ${\sigma }_8=0.58$. This figure shows the consistency with Fig. 1 in Ref. [44].

Figure 2

$\lambda$, $\mu$, $\nu$, ${\lambda }_{\theta }$, ${\mu }_{\theta }$, ${\nu }_{\theta }$ as functions of the scale factor $a$. In each panel, the blue dash-dotted curve is the $\mathrm{\Lambda }\mathrm{CDM}$ model, and the red dotted curve, the yellow dashed curve, and the green thick solid curve show the KGB model with $n=1$, 2, and 5, respectively.

Figure 3

Comparison of $P_{\delta \delta }(k)$ (thick solid curve), $P_{\delta \theta }(k)$ (dashed curve), $P_{\theta \theta }(k)$ (dotted curve), and the linear power spectrum $P_{\mathrm{lin}}=D_{+}^2(t)P_L(k)$ (thin solid curve), at the redshift $z=0$. The upper left panel is the $\mathrm{\Lambda }\mathrm{CDM}$ model, and the other panels show the KGB model with $n=1$, 2, and 3, as described in each panel. Here we adopted ${\mathrm{\Omega }}_0=0.3$, ${\mathrm{\Omega }}_b=0.044$, $h=0.7$, $n_s=0.96$, and ${\sigma }_8=0.8$. In this figure, the same linear power spectrum is adopted for all the models.

Figure 4

Relative deviation of the power spectra $P_{\delta \delta }(k)$ (top panel), $P_{\delta \theta }(k)$ (middle panel), and $P_{\theta \theta }(k)$ (bottom panel), under the kinetic gravity braiding model with $n=1$ (red dotted curve), $n=2$ (yellow dashed curve), and $n=5$ (green thick curve), which are divided by those under the $\mathrm{\Lambda }\mathrm{CDM}$ model. The panels show the snapshot at the redshift $z=0$.